The Kohn-Laplace equation on abstract CR manifolds: Global regularity

Abstract:

Let $M$ be a compact, pseudoconvex-oriented, $(2n+1)$-dimensional, abstract CR manifold of hypersurface type, $n\geq 2$. We prove the following:

(i) If $M$ admits a strictly CR-plurisubharmonic function on $(0,q_0)$-forms, then the complex Green operator $G_q$ exists and is continuous on $L^2_{0,q}(M)$ for degrees $q_0\le q\le n-q_0$. In the case that $q_0=1$, we also establish continuity for $G_0$ and $G_n$. Additionally, the $\bar {\partial }_{b}$-equation on $M$ can be solved in $C^\infty (M)$.

(ii) If $M$ satisfies “a weak compactness property” on $(0,q_0)$-forms, then $G_q$ is a continuous operator on $H^s_{0,q}(M)$ and is therefore globally regular on $M$ for degrees $q_0\le q\le n-q_0$; and also for the top degrees $q=0$ and $q=n$ in the case $q_0=1$.

We also introduce the notion of a “plurisubharmonic CR manifold” and show that it generalizes the notion of “plurisubharmonic defining function” for a domain in $\mathbb {C}^N$ and implies that $M$ satisfies the weak compactness property.